By Alonso Peña
This ebook will introduce you to the main mathematical types used to cost monetary derivatives, in addition to the implementation of major numerical types used to resolve them. specifically, fairness, forex, rates of interest, and credits derivatives are mentioned. within the first a part of the publication, the most mathematical types utilized in the realm of economic derivatives are mentioned. subsequent, the numerical equipment used to resolve the mathematical versions are offered. ultimately, either the mathematical types and the numerical equipment are used to unravel a few concrete difficulties in fairness, currency, rate of interest, and credits derivatives.
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Extra info for Advanced Quantitative Finance with C++
Like in MC simulation, time is discretized in steps dt from t=0 to t=T. From one step tp, the next price of the underlying can either go up or down by a factor u or d as shown in the following formulas: At t = 0 : S0 At t = t1 : uS0 or dS0 At t = t2 : u 2 S0 , udS0 , d 2 S0 Thus, the values of the tree at maturity are as follows: ST1 = u 2 S 0 , ST2 = udS 0 , ST3 = d 2 S 0 In general case, we proceed in a similar manner until we arrive at the maturity T, and we have N+1 values for the variable S.
The third phase is the backward phase. In this final phase, we take the values of the payoff at maturity and proceed in a backward manner. We move from the last node to the previous nodes, by computing the option value as the discounted expected payoff in the previous nodes using the weighted probabilities, as follows: VTk 1 = exp( r t) pVTk + (1 p)VTk In our case, in the second step, the equations are as follows: V21 = exp( r t) pV31 + (1 p)V32 V22 = exp( r t) pV32 + (1 p)V33 And, in the first step, the equation is as follows: V11 = exp( r t) pV21 + (1 p)V22 The premium of the derivative, the option price, is the value S 9.
This is the origin of the name of finite differences. For more details, the reader can refer to The Mathematics of Financial Derivatives: A Student Introduction. Consider the following illustration where a continuous function f(X) and the first derivative of the function is defined as follows: I [ GI 'I | G[ '[ IL L IL [L L [L IL L IL '[ The preceding function is also known as the slope, which is the ratio between the growth (or decrease) in the function with respect to the step size dx.